#122157

order	512
nilpotency class	3
isoclinism	isoclinic class of #122155
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,10)(4,6)(5,9)(7,12)(8,14)(11,15)(13,16)(17,18,21,23,22,26,20,25)(19,27,28,31,29,24,30,32);(2,6,9,8)(7,16,10,15)(18,23)(20,21)(24,31)(25,26)(27,32)(28,30);(1,3)(2,7,9,10)(4,13)(5,12)(6,15,8,16)(11,14)(17,19)(18,24)(20,28)(21,30)(22,29)(23,31)(25,32)(26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,g^b=gi,d^c=dh,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29353);